# Definite Integral $\int_{\frac{\pi}{12}}^{\frac{\pi}{4}}\frac{8\cos2x}{(\tan x+\cot x)^3}\,dx$. $$\int_{\frac{\pi}{12}}^{\frac{\pi}{4}}\dfrac{8\cos2x}{(\tan x+\cot x)^3}\,dx$$

The problem is attached as a photograph!

so I started this, by taking cot(x) as 1/tan(x) and then after simplifying, I got, 8{2sin³(x)cos^5(x)- sin³(x)cos³(x)}

After converting these to Sin(2x) I got,

2cos²x(sin2x)³ - (sin2x)³

Taking sin³2x common, we get,

Sin³2x(cos2x)

Taking sin2x = t Hence (2cos2x)dx=dt

I get (1/2)* integral of (t³)dt

Which gives me t⁴/8 as the ans

Now substitution part. It's no where near to the options. MI going wrong somewhere? Or the whole method itself is not correct/cannot be put to use here?

I have just started studying integration. So pls help me out. PS: sorry for making this all so boring!

• i can't see the image – Dr. Sonnhard Graubner Oct 3 '17 at 10:03
• you should start with learning LaTex – user392395 Oct 3 '17 at 10:04
• There is a small "1" digit there – Arnav_05 Oct 3 '17 at 10:04
• @Fightclub1995 Ok I want to but idk where to start from – Arnav_05 Oct 3 '17 at 10:05
• @Dr.SonnhardGraubner see if it is working – Arnav_05 Oct 3 '17 at 10:06

## 2 Answers

$\tan x+\cot x=\dfrac{2}{\sin2x}$ so $$I=\int_{\frac{\pi}{12}}^{\frac{\pi}{4}} 8\cos2x\sin^32x\,dx=\dfrac18\sin^42x\Big|_{\frac{\pi}{12}}^{\frac{\pi}{4}} = \color{blue}{\dfrac{15}{128}}$$

$$\frac{8\cos2x}{(\tan{x}+\cot{x})^3}=8\cos2x\sin^3x\cos^3x=\cos2x\sin^32x=$$ $$=\frac{\cos2x(3\sin2x-\sin6x)}{4}=\frac{1}{8}(2\sin4x-\sin8x).$$ Also, we have $$\cos2x\sin^32xdx=\frac{1}{2}\sin^32xd(\sin2x).$$ I hope now it's clear.

• And the main mistake was of not changing the limits!!! – Arnav_05 Oct 3 '17 at 10:15
• @user478518 Now you can easy calculate the integral. – Michael Rozenberg Oct 3 '17 at 10:17
• @user478518 You are welcome! – Michael Rozenberg Oct 3 '17 at 10:19
• @MichaelRozenberg. Hi, just wondering, why aren't you just immediately integrating from $8\cos2x\sin^32x$ ? Derivative of sine is cosine after all... – imranfat Oct 4 '17 at 1:36
• @imranfat It was typo. Thank you! – Michael Rozenberg Oct 4 '17 at 2:15